We have adopted a size of 60 for the envelope of L1157-mm, a value in good agreement with the size of the envelope
observed in N\(_2\) H\(^+\)  [1]}. In practice, this value allowed us to best simultaneously reproduce the emission of the various transitions of a given molecular species under the assumption of one single excitation temperature.
We have adopted a total gas column density of \(2\times 10^{22}\hbox{\hspace{1.99997pt}cm$^{-2}$}\) , based on the H\({}_2\)  column density maps of the region obtained in the Herschel Gould Belt Survey (HGBS) of nearby star-forming molecular clouds [2]}. We note that this H\({}_2\)  column density is about a factor of 3 higher
than the determination obtained by [3]}. The latter value (\(6\times 10^{21}\hbox{\hspace{1.99997pt}cm$^{-2}$}\) ) was derived from the \(^{13}\) CO \(J\) =1–0 line in the ASAI spectrum of the protostar, adopting a canonical abundance of \(1.6\times 10^{-6}\) . As a matter of fact, the low gas kinetic temperature (\(\sim 10\hbox{\hspace{1.99997pt}K}\) ) and the relatively high gas density (\(\sim \)  a few \(10^4\hbox{\hspace{1.99997pt}cm$^{-3}$}\) ) of the envelope are such that CO is most likely depleted at large scale in the envelope by a factor of a few, as is commonly measured in cold cores [4]}. Both column density values can be reconciled if CO is moderately depleted by a factor of about 3–4 over the envelope. We have adopted a linewidth of \(\sim 1\hbox{\hspace{1.99997pt}km\hspace{1.99997pt}s$^{-1}$}\)  for the line profile modelling in L1157-mm, in agreement with the results of our gaussian fits to the line spectra (see Figs. REF –REF  and Table 2).
